1. Mixtures and Hierarchical Mixtures of Cox Experts The goal during the present grant period will be to study and extend a mixture model which combines features of the usual Cox proportional hazards model with those of mixtures-of-experts. The first subaim will be to compare the mixture of Cox experts approach to spline-based methods. The second subaim is to develop and study a Markov chain Monte Carlo approach to inference for the mixture of Cox experts. The third subaim will be to develop, study and apply a hierarchical mixture of Cox experts architecture. 2. Semiparametric Bayesian Inference for Regression models Seifu, Severini and Tanner (1997) present a model for Bayesian inference in a linear model with independent and identically distributed errors that does not require the specification of parametric family of densities for the error distribution. The focus of this specific aim is to extend this work to the censored regression case. Both the one-sample and regression cases will be considered. This new methodology for censored data problem will be examined via stimulation studies and validated using real data. 3. Approximate Monte Carlo Conditional Inference Kolassa and Tanner (1997) present an algorithm for approximate Frequentist conditional inference on two or more parameters. The method makes use of the double saddle point approximations of Skovgaard (1987) to the conditional cumulative distribution function of a sufficient statistic given the remaining sufficient statistics. This approximation is then used in conjunction with non-iterative simulation methods to generate a sample from the distribution that approximates the joint distribution of the sufficient statistics associated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters. The focus of this specific aim is to further study and apply the algorithm of Kolassa and Tanner (1997) to general problems in biostatistics. The algorithm will be applied to situations such as general hierarchical log-linear models for multiway contingency tables and logistic regression. The algorithm will be compared via simulation and on real data sets to alternative methods for exact conditional inference. The methodology will be improved by replacing the Skovgaard approximation with a higher-order approximation due to Kolassa (1996). This methodology will also be applied to a variety of models in the generalized linear model family for which no competing exact or higher-order methods are available.